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Sequence of integral of ratio between two powers of functions

Source: Romanian District Olympiad 2002, Grade XII, Problem 2

October 7, 2018
functionSequencesIntegralcalculusintegrationratio

Problem Statement

Let be two distinct continuous functions f,g:[0,1]⟶(0,∞) f,g:[0,1]\longrightarrow (0,\infty ) corelated by the equality ∫01f(x)dx=∫01g(x)dx, \int_0^1 f(x)dx =\int_0^1 g(x)dx , and define the sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} as xn=∫01(f(x))n+1(g(x))ndx. x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx .
a) Show that ∞=lim⁡n→∞xn. \infty =\lim_{n\to\infty} x_n. b) Demonstrate that the sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} is monotone.
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